I'm not an expert myself, but I've done a lot of reading. I don't have a specific source at hand, but this is the consensus I've gotten from any interviews of vaccinologists or epidemiologists I've read or heard. Essentially that any side effects from vaccinations show up within a few weeks—that there just isn't a mechanism for them to appear years down the line if they haven't already been seen sooner, for the reason I described.

Now again it is possible that side effects could still be discovered in patients with complicating factors that weren't represented in the trials (like pregnancy or other known or unknown pre-existing conditions). Or just because the effects were too rare to show up significantly in the trials. But again these would be expected to show up quickly as widespread vaccination begins, as did the few severe allergic reactions that have occurred.

Of course as with anything, especially as charged an issue as vaccines, if one goes looking for it one can find plenty of purported evidence that long-term side-effects (by which I mean here side effects that don't show up until long after the vaccine is taken) are possible or even common. But the expert consensus based on the totality of evidence appears to be that this is not a serious concern.